HI 



MATHEMATICS. PLANE GEOMKTKY. [BOOK in. PROP. IT. xrv. 



I. than M D,t and that M K - MO, /. the re- 

 maiuder K D U greater than the remainder G D ; that is, 

 G D U les tli.-ui K D, any other line from D to the con- 

 rex part of the circumference, .'. D G i* the least Hnr. 

 1 >, ..> M I. : then because M LD U a triangle, and that 

 M K, D K are drawn from the extremities of a side to a 

 point K. within the triangle, .'. 

 M K + D K are less than M L 

 III. + D L :* but M K - 

 M L, .'. the remainder D K is 

 leu than the remainder D L ; 

 that U, a line nearer to the least 

 it less than one more remote. 



Also, there can be drawn ttco, 

 but only two, equal straight lines 

 from D to the circumference, one 

 on each side of DA, the line 

 through the centre. For draw any 

 straight line DK, and at M make 

 + M i. the angle DMB-DM K ;t 

 ami draw D B. Then because 

 M K = M B, the two M K, M D 

 are = the two M B, M D, each to 

 each ; and the angle K M 1 > 



Const. BMD,' .'. DK = 

 + 41. D B,t in that tico equal lines can be drawn, and 

 the demonstration is the same whether D K be drawn to 

 the convex or to the concave part of the circumference. 

 But, besides D B, no straight line drawn from D to the 

 circumference, can be = DK. For, if there can, let it 

 be D N. Then because D K = D N, and D K = D B, .'. 

 D N = D B ; but one of these must be nearer to that 

 through the centre than the other, .'. a line nearer to 

 that through the centre is = one more remote, which has 

 been proved to be impossible ; .". if any point, &~ 

 Q. E D. 



PROPOSITION IX. THEOREM. 



If any point be taken, from which there may be drawn 

 more than two equal straight lines to the circumference 

 of a circle, that puint is the centre of tlie circle. 



For if the point were not the centre, only two equal 

 straight lines could be drawn from it to the circumfer- 

 7 * 8 ill. ence ;* .' . if any point, <tc. 



PROPOSITION X. THEOREM. 



One circumference of a circle cannot cut another in more 

 points than two. 



For suppose the circumference F A B C to cut the cir- 

 cumference D E G F in more than two points, viz. , in 

 B, G, F. Take the centre K, of the circle F A B C,* 

 and draw K B, K G, K F. Then because K 

 is the centre of the circle FAB, 

 these lines are all equal; so that 

 from a point K, to the other 

 circumference D E G F, more 

 than two equal straight lines 

 are drawn, viz. , the three K B, 

 KG, KF, .'. K is the centre 

 + 9 in. of the circle DEGF.f 

 . . the same point is the centre 

 of two circles that cut one 



s ill. another, which is impossible ;* . . one circum- 

 ference, <tc. Q. E. D. 



PROPOSITION XL THEOREM. 



If one CM-CM (A D E) touch another ^A B C) internally in a 

 potnt (A), the straight line which joins their centres being 

 prolonged, shall pass through the point of contact (A). 



Let F be the centre of ABC, and O, the centre o: 

 A D E ; F G, when prolonged, shall pass through A. 



For if not, let F G, prolonged, cut the circumferences 

 in D and H. Draw A F, A G : then, because two sides 

 of a triangle are together greater than the third side 



Ml. FO + OAare greater than FA : but FA- 



i in. 



FH, .-. FG + GA are greater 

 than FH. Take away the com- 

 mon part F G, . . G A is greater 

 than G H : but G A - G D,t .'. 

 + Def. u I. G D is greater than 

 G H, which is impossible, even 

 could D, H coincide, . . FG being 

 'C prolonged, cannot but pass through 

 the point A ; . . if one circle, <ta 

 Q. E. D. 



PROPOSITION XII. THEOREM. 



If two circles (A B C, A D E) touch each other externally 

 in a point (A), the straight line which wins tiieir centres 

 shall pass through the point of contact (A). 



Let F be the centre of A B C, and G that of A D E : 

 F G shall pass through A. 



For if not, let F G cut the circumferences in C, D, and 

 draw F A, G A. Then because F is the centre of A B C, 

 F A = F C and because G is the centre of A D E, G A 



G D, .-. F A + 



the whole F G 

 not lets than F A 

 G A, even though 

 C, D be supposed to co- 

 incide ; but a side F G, 

 of the triangle A F G, u 

 less than the other two 



SOI. sides,* which is impossible, .-. FG cannot 

 pass otherwise tlian through the point of contact A ; 

 .'. if two circles, &c, Q.E. D. 



PROPOSITION XIII. THEOREM. 



One circle cannot touch another in more than one point, 

 whether it touch it on the inside or on the outside. 



For, if it be supposed possible, let the circle E B F 

 touch the circle A B C in two points ; and first on the 

 inside, in the points B, D. Join B, D, and draw G H, 



10 & 11 1. bisecting BD at right angles.* Then because 

 the points B, 



D are in the 

 circumference 

 of each of the 

 circles, B D 

 falls within 

 each of them, 



2 III. 



their eentrts 

 are in G H, 



1 III. Cor. which bisects B D at right angles,* .'. G H 

 passes through the point of contact B, and through the 



+ u in. point of contact D :t but the points B, D are 

 without the straight line G H, which is absurd ; .'. one 

 circle cannot touch another on the inside in more than one 

 point. 



Nor can two circles touch on the outside in more than 

 one point. For suppose the circles ACK, ABC to 

 touch in two points, A, C. Join A, C : then because 

 A, C are in the circumference ACK, AC, which joins 

 them, falls within the circle ACK* 



2 in. But the circle A C K ia 

 + Hyp. without the circle ABC,t 



.'. A C is without this last circle ; 

 but, because A, C are in the cir- 

 cumference ABC, AC must be 



2 ill. within the same circle,* 

 which is absurd, .'. one circle cannot 

 touch another on the outside in more 

 than one point ; and it has been 

 shown that they cannot touch on 

 the inside in more than one point, 

 .'. one circle, &c. Q. E. D. 



PROPOSITION XIV. THEOREM. 

 Equal chords (A B, C D) in a circle are ej'ually distant 



